## Abstract

The importance of phonons and their interactions in bulk materials is well known to those working in the fields of solid‐state physics, solid‐state electronics, optoelectronics, heat transport, quantum electronic, and superconductivity. Phonons in nanostructures may act as a guide to research on dimensionally confined phonons and lead to phonon effects in nanostructures and phonon engineering. In this chapter, we introduce phonons in zinc blende and wurtzite nanocrystals. First, the basic structure of zinc blende and wurtzite is described. Then, phase transformation between zinc blende and wurtzite is presented. The linear chain model of a one‐dimensional diatomic crystal and macroscopic models are also discussed. Basic properties of phonons in wurtzite structure will be considered as well as Raman mode in zinc blende and wurtzite structure. Finally, phonons in ZnSe, Ge, SnS2, MoS2, and Cu2ZnSnS4 nanocrystals are discussed on the basis of the above theory.

### Keywords

- phonons
- zinc blende
- wurtzite
- Raman spectroscopy
- molecular vibration

## 1. Zinc blende and wurtzite structure

Crystals with cubic/hexagonal structure are of major importance in the fields of electronics and optoelectronics. Zinc blende is typical face‐centered cubic structure, such as Si, Ge, GaAs, and ZnSe. Wurtzite is typical hexagonal close packed structure, such as GaN and ZnSe. In particular, II–VI or III–V group semiconductor nanowires always coexist two structures, one cubic form with zinc blend (ZB) and another hexagonal form with wurtzite (WZ) structure. Sometimes, this coexistence between zinc blende and wurtzite structure leads to form twinning crystal during the phase transformation between zinc blende and wurtzite [1, 2].

### 1.1. Basic structure of zinc blende and wurtzite

The crystal structure of zinc selenide in the zinc blende structures is shown in Figure 1, which is regarded as two face‐centered cubic (fcc) lattices displaced relative to each other by a vector

Figure 2 is wurtzite structure of zinc selenium. Close‐packed planes of wurtzite are {0001} along <0001>, and the stacking is …ABABA…. Adjacent plane spacing is c/2. Wurtzite structures have four atoms per unit cell. In zinc blende, the bonding is tetrahedral. The wurtzite structure may be generated from zinc blende by rotating adjacent tetrahedra about their common bonding axis by an angle of 60° with respect to each other.

### 1.2. Phase transformation between zinc blende and wurtzite

Research into controlling nanowire crystal structure has intensified. Several reports address the diameter dependency of nanowire crystal structure, with smaller diameter nanowires tending toward a WZ phase and larger diameter nanowires tending toward a ZB phase. Allowing for ZnSe, two phases, zinc blende (ZB) and wurtzite (WZ), exist, and the (111) faces of ZB phase are indistinguishable from and match up with the (001) faces of WZ phase, the subtle structural differences of which lead to the attendant small difference in the internal energies (∼5.3 meV/atom for ZnSe). The WZ‐ZB phase transformation is considered to be caused by the crystal plane slip. Take the formation of ZnSe longitudinal twinning nanowires, for example [3]. Structurally, the (001) planes of WZ and the (111) planes of ZB are their corresponding close packing planes. ABAB stacking for WZ and ABCABC stacking for ZB are shown in Figure 3a and b, respectively. It was noteworthy that the arrangement of atoms in A/B packing planes was different in WZ phase. So the phase transition could not be realized until the smaller Zn atoms moved to the interspaces provided by three neighboring bigger Se atoms, within the plane B. In this case, the new layers B' were obtained, and then, the slip occurs between neighboring planes A and B’ by

Generally, there are three equivalent directions to realized the slip, which are <120>, <

## 2. Linear‐chain model and macroscopic models

To the simple double lattice, lattice vibration can be described by the one‐dimensional diatomic model. The linear‐chain model of a diatomic crystal is based upon a system of two atoms with masses, *m* and *M*, placed along a one‐dimensional chain as depicted in Figure 4. The separation between the atomics is “*a*”, and the vibration in the vicinity of their equilibrium position is treated as the simple harmonic vibration. The properties of optical phonon can be described based on the macroscopic fields. It is the model based on the Huang and Maxwell equations, which has great utility in describing the phonons in the uniaxial crystals such as wurtzite crystals.

### 2.1. Polar semiconductors

Polar semiconductor is the crystal that consists of different ions. In polar semiconductor, the lattice vibration is associated with the electric dipole moment and electric field generation. Assume that the vibration frequency is *ω*, wave vector is

Solve the simultaneous formula (2‐1) and Maxwell equations can obtain,

To longitudinal polarity lattice mode,

To transverse polarity lattice mode,

As was apparent above, polar optical phonon vibrations produce electric fields and electric polarization fields that may be described in terms of Maxwell's equations and the driven‐oscillator equations. Assume that the mass of the ions are *k*,

where

where

The lattice vibration is associated with the electric dipole moment generation, which can be described as follows,

where

Replace the value of

Then, take formula (2‐7) and (2‐9) into (2‐10),

where

formula (2‐11) and (2‐12) are the Huang equations, which are the basic equations of describing the vibrations of long wave in the polar crystals. From the formula (2‐14) and (2‐16), one can find that,

When the system is under the high‐frequency electric field, formula (2‐12) reduces to

For

Compute the curl of formula (2‐11) and solve the simultaneous equations of (2‐12) and electrostatic equations

When the system is under the static electric field,

Take formula (2‐21) into (2‐12),

Replace the electrostatic equation,

And take formula (2‐23) and (2‐20) into (2‐22),

Solve the simultaneous equations of (2‐17) and (2‐24) can obtain

Solve two simultaneous Maxwell and Huang equations,

Assume the solution forms are

Take (2‐28) into the Huang and Maxwell equations,

To the longitudinal wave,

Take (2‐19) (2‐20) (2‐25) (2‐26) into (2‐31)

Equation (2‐23) is the dispersion relations of longitudinal wave, which is commonly called Lyddane‐Sachs‐Teller (LST) relationship. LST relation indicates that the frequency of longitudinal wave is a constant and independent on the wave vector.

Similarly, to the transverse wave,

Replace the values of *A*, *B*, *C*, and *D* into (2‐33),

Equation (2‐34) is the dispersion relations of transverse wave. One can find that the frequency of transverse is dependent on the value of wave vector

### 2.2. Dispersion relations

One‐dimensional diatomic model can be regarded as the simple double lattice. In the simple linear chain model, it is assumed that only nearest neighbors are coupled, and that the interaction between these atoms is described by Hooke's law; the spring constant *α* is taken to be that of a harmonic oscillator. Thus, the kinematical equations are established,

where *m* and *M* are the mass of the adjacent atoms *n*-1, 2*n*, 2*n* + 1, and 2*n* + 2, respectively.

where

Eliminating *A*_{1} and *A*_{2},

The relationship between frequency and wave vector is commonly called dispersion relation [5].

## 3. Basic properties of phonons in wurtzite structure

In this section, we discuss the phonon effects in wurtzite structure. The crystalline structure of a wurtzite material is depicted in Figure 2. There are four atoms in the unit cell. Thus, the total number of optical modes in the long‐wavelength limit is nine: three longitudinal optic (LO) and six transverse optic (TO). In these optical modes, there are only three polar optical vibration modes. According to the group theory, the wurtzite crystal structure belongs to the space group

Due to the anisotropy of wurtzite structure, the vibrational frequency of oscillates parallel and perpendicular to the optical axis is denoted by

where

When the wave vector is parallel to the optical axis,

which is the same as formula (3‐1). When the wave vector is perpendicular to the optical axis,

Formula (3‐4) indicates that the extraordinary wave is transverse wave when the wave vector is perpendicular to the optical axis.

When

and

Formula (3‐5) indicates that frequency of ordinary phonon is independent on the wave vector *q*. Formula (3‐6) indicates that the frequency of extraordinary phonon is dependent on the orientation of the wave vector, but independent on its value.

It is most convenient to divide uniaxial crystals into two categories: (a) the electrostatic forces dominate over the anisotropy of the interatomic forces and (b) the short‐range interatomic forces are much greater than the electrostatic forces. It has been turned out that crystals with the wurtzite symmetry fall into the first category. In this case,

thus,

and

## 4. Raman mode in zinc blende and wurtzite structure

Raman spectroscopy is a non‐destructive technical tool used to gain information about the phonon behavior of the crystal lattice through the frequency shift of the inelastically scattered light from the near surface of the sample. It is well known that different crystal phases have different vibrational behaviors, so the measured Raman shifts of different phases are mostly unique and can be seen as fingerprints for the respective phases. This provides the possibility of detecting different phases in a sample. It has been developed to be a versatile tool for the characterization of semiconductors leading to detailed information on crystal structure, phonon dispersion, electronic states, composition, strain, and so on of semiconductor nanostructures.

In a zinc blende structure, the space group of the cubic unit cell is *F*43*m*(

The wurtzite crystal structure belongs to the space group *A*_{1}, 2*B*_{1}, *E*_{1,} and 2*E*_{2}. The *A*_{1} and *E*_{1} modes and the two *E*_{2} modes are Raman active, whereas the *B* modes are silent. The *A* and *E* modes are polar, resulting in a splitting of the LO and the TO modes [6].

## 5. Phonons in ZnSe, Ge, SnS_{2}, MoS_{2}, and Cu_{2}ZnSnS_{4} nanocrystals

In addition to the attached references, this chapter is primarily written on the basis of our research works. Here, we select ZnSe, Ge nanowires and CdSe/Ge‐based nanowire heterostructures, two‐dimensional semiconductors SnS_{2} and MoS_{2}, and candidate absorber materials of thin‐film solar cells Cu_{2}ZnSnS_{4}. These examples will help us to understand the phonons behaviors in nanostructures.

It is well known that ZnSe has two structures: cubic zinc blende (ZB) and hexagonal wurtzite (WZ) due to the difference of the stacking sequence of successive layers, whereas Ge has diamond structure. SnS_{2} and MoS_{2} belong to the wide family of compounds with layered structures. SnS_{2} crystal is isostructural to the hexagonal CdI_{2}‐type structure. MoS_{2} usually consists of a mixture of two major polytypes of similar structure, 2H (hexagonal) and 3R (rhombohedral), with the former being more abundant. As for quaternary Cu_{2}ZnSnS_{4} (CZTS), the parent binary II‐VI semiconductors adopt the cubic zinc blende structure, and the ternary I‐III‐VI_{2} compounds can be generated by mutating the group II atoms into pairs of group I and III atoms. The quaternary CZTS materials are formed by replacing the two In (III) atoms with Zn (II) and Sn (IV), respectively (see Figure 5).

We use Raman spectroscopy to identify crystal structure of ZnSe one‐dimensional material (Figure 6). In sample S3, the Raman peaks at 204 and 251 cm^{-1} are attributed to the scatterings of the transverse optic (TO) and longitudinal optic (LO) phonon modes of ZnSe, respectively. A strong peak at 232 cm^{-1}, between the TO and LO phonons, is thought to be surface mode. The Raman peak at ∼176 cm^{-1} is attributed to the hexagonal phase *E*_{1}(TO) mode of ZnSe, which is inhibited in Raman spectrum (RS) of ZB ZnSe. Compared with S3, Raman peaks at 205.6 (TO mode) and 252 cm^{-1} (LO mode) of S1 show tiny blue‐shift. However, in S1, there is no Raman peak corresponding to the surface mode, as well as *E*_{l} (TO) mode, which is suppressed in the ZB phase. This indicates the existence of ZB phase in S1. Thus, structure of the sample can be shown through RS, and we got S1‐ZB phase, S3‐WZ, S2 the coexist of ZB and WZ [7] (cm^{-1}).

Figure 7 shows the room temperature RS of CdSe/Ge‐based nanowires. The LO mode of Ge in CdSe‐Ge (or CdSe‐Ge‐CdSe), ‐CdSe‐Ge core/polycrystalline Ge sheath, and ‐Ge‐GeSe heterostructural nanowires has a downshift by 8, 5, and 2 cm^{-1} in comparison with that of the bulk counterpart Ge (299 cm^{-1}), respectively. With regard to the microstructure of heterostructural nanowires, the downshift of the LO mode may be caused by tensile stress, which affects the Raman line by a downshift. And the different shift scales are attracted by the different sizes of the Ge subnanowires and Ge nanocrystalline [8].

The individual layer in SnS_{2} is known as an S‐Sn‐S sandwich bonded unit. Each Sn atom is octahedrally coordinated with six nearest neighbor sulfur atoms, while each S atom is nested at the top of a triangle of Sn atoms. The sandwich layers in the elementary cell occur along the *c* axis and bonded together by Vander Waals forces. The normal modes of vibration in SnS_{2} are given by the irreducible representations of the *D*_{3d} point group at the center of the Brillouin zone: Γ = *A*_{lg} + *E*_{g} + 2*A*_{2u} + 2*E*_{2u}. Two Raman‐active modes (*A*_{1g} and *E*_{g}) and two IR‐active modes (*A*_{2u} and *E*_{u}) are found. In view of the existence of an inversion center, the IR‐ and Raman‐active modes are mutually exclusive. On the other hand, six atoms in the unit cell of SnS_{2} extend over two sandwich layers. Eighteen normal vibration modes can be represented by the following irreducible form: Γ = 3*A*_{1} + 3*B*_{1} + 3*E*_{1} + 3*E*_{2}. Based on the analysis above, there are six modes, which are both IR‐ and Raman‐active, belonging to *A*_{1} and *E*_{l}, and three Raman‐active modes belonging to *E*_{2}. The *B*_{1} modes are silent, while the three acoustic modes belong to *A*_{1} and *E*_{1} [9].

The RS of β‐SnS_{2} nanocrystal is illustrated in our former work [10]. The spectra show one first‐order peak at 312 cm^{-1} that corresponding to *A*_{1g} mode. The RS of as‐prepared SnS_{2} shows a slight redshift in comparison with that of bulk materials (peak at 317 cm^{-1}). The redshift of phonon peaks is due to spatial confinement of phonon modes. The first‐order *E*_{g} mode (peak at 208 cm^{-1}) cannot be observed, which likely results from a nanosize effect. A wide peak between 450 and 750 cm^{-1}, which only observed in the bulk materials at lower temperature, may be attributed to second‐order effects.

The phonon dispersion of single‐layer MoS_{2} has three acoustic and six optical branches derivatized from the nine vibrational modes at the Γ point. The three acoustic branches are the in‐plane longitudinal acoustic (LA), the transverse acoustic (TA), and the out‐of‐plane acoustic (ZA) modes. The six optical branches are two in‐plane longitudinal optical (LO_{1} and LO_{2}), two in‐plane transverse optical (TO_{1} and TO_{2}), and two out‐of‐plane optical (ZO_{1} and ZO_{2}) branches.

For 2L and bulk MoS_{2}, there are 18 phonon branches, which are split from nine phonon branches in 1LMoS_{2}. The phonon dispersions of 1L and bulk MoS_{2} are very similar, except for the three new branches below 100 cm^{-1} in bulk because of interlayer vibrations. There are similar optical phonon dispersion curves for 1L, 2L, and bulk MoS_{2} because of the weak Vander Waals interlayer interactions in 2L and bulk MoS_{2} [11].

Raman spectroscopy is also used to accurately identify the layer number of MoS_{2}. The frequency difference between out‐of‐plane *A*_{1g} and in‐plane *E*_{2g}^{1} mode of MoS_{2} is denoted as ._{2}, ^{-1} to 25.5 cm^{-1}. In our work [12], two strong peak at ∼379 cm^{-1} and ∼402 cm^{-1} can be assigned as in‐plane *E*_{2g}^{1} mode and out‐of‐plane *A*_{1g} mode of MoS_{2}, respectively, which has a redshift in comparison with that of the bulk MoS_{2}. The ^{-1}, indicating that the as‐grown MoS_{2} contains tri‐layer MoS_{2.}

The phonon dispersion and density‐of‐states curves along the principal symmetry directions of kesterite CZTS were calculated using a density functional theory by Khare et al. [13]. The phonon states around 50–160 cm^{-1} are mainly composed of vibrations of the three metal cations with some contribution from the sulfur anions. The phonon states around 250–300 cm^{-1} are mainly composed of vibrations of the Zn cations and S anions with some contribution from the Cu cations. The phonon states from 310 to 340 cm^{-1} are mainly a result of vibrations of S anions, whereas those from 340 to 370 cm^{-1} are composed of the vibrations of Sn cations and S anions.

To more exactly confirm secondary phases in Cu_{2}‐II‐IV‐VI_{4} semiconductors, Raman scattering studies have been extensively performed. From the vibrational point of view, the zone‐center phonon representation of the kesterite structure space group *I**A* + 6*B* + 6*E*_{1} + 6*E*_{2}, where 12*B*, *E*_{1}, and *E*_{2} modes are infrared active, whereas 15*A*, *B*, *E*_{1,} and *E*_{2} modes are Raman active. According to our work [14], the single peak at about 328 cm^{-1} of Raman spectrum of the as‐prepared CZTS nanocrystals can be assigned to breathing mode of sulfur atoms around metal ions in CZTS. Moreover, Raman spectrum of CZTS has about 8 cm^{-1} redshifts compared with that of the responding bulk counterpart which may be due to a smaller size effect.

In our work of fabrication of Cu_{2}ZnSnS_{x}Se_{4-x} solid solution nanocrystallines [15], RS revealed that vibrating modes were modulated by *x*‐values. The peak position of 170, 189, and 229 cm^{-1} shifted to higher frequency with increasing *x*‐value in CZTSSe, respectively. Those peaks completely disappeared when *x* = 4. Moreover, a wide peak located at about 330 cm^{-1} appeared when *x* > 0 and the relative intensity increased with increasing *x*‐value. Such results indicate that Se elements were gradually replaced by S elements in CZTSSe solid solution system.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11174049 and 61376017.